scholarly journals Numerical approach for solving fractional relaxation–oscillation equation

2013 ◽  
Vol 37 (8) ◽  
pp. 5927-5937 ◽  
Author(s):  
Mustafa Gülsu ◽  
Yalçın Öztürk ◽  
Ayşe Anapalı
Keyword(s):  
Relaxation Oscillation ◽  
Numerical Approach ◽  
Fractional Relaxation ◽  
Oscillation Equation

Exact Analytical Solution of Fractional Relaxation-Oscillation Equation by Adomian Decomposition and He’s Variational Technique

2009 ◽  
Author(s):  
K. C. Basak ◽  
P. C. Ray ◽  
R. K. Bera
Keyword(s):  
Exact Solution ◽  
Analytical Solution ◽  
Fractional Order ◽  
Decomposition Method ◽  
Exact Analytical Solution ◽  
Relaxation Oscillation ◽  
Variational Technique ◽  
Adomian Decomposition ◽  
Fractional Relaxation ◽  
Oscillation Equation

Exact solution of linear fractional relaxation-oscillation equation is obtained by the decomposition method of Adomian and also by He’s variational method for fractional order α, for 1 < α ≤ 2. Surface plots of the above solution are drawn for different values of fractional order α and time t. Amplitude of the oscillation increases with α but it decreases as time increases.

Generalized wavelet collocation method for solving fractional relaxation–oscillation equation arising in fluid mechanics

2017 ◽  
Vol 06 (02) ◽  
pp. 1750016
Author(s):  
Firdous A. Shah ◽  
R. Abass
Keyword(s):  
Fluid Mechanics ◽  
Relaxation Oscillation ◽  
Operational Matrix ◽  
Haar Wavelet ◽  
Test Problems ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Specific Test ◽  
Fractional Relaxation ◽  
Oscillation Equation

In this paper, a generalized wavelet collocation operational matrix method based on Haar wavelets is proposed to solve fractional relaxation–oscillation equation arising in fluid mechanics. Contrary to wavelet operational methods accessible in the literature, we derive an explicit form for the Haar wavelet operational matrices of fractional order integration without using the block pulse functions. The properties of the Haar wavelet expansions together with operational matrix of integration are utilized to convert the problems into systems of algebraic equations with unknown coefficients. The performance of the numerical scheme is assessed and tested on specific test problems and the comparisons are given with other methods existing in the recent literature. The numerical outcomes indicate that the method yields highly accurate results and is computationally more efficient than the existing ones.

scholarly journals Fractional Relaxation and Fractional Oscillation Models Involving Erdélyi-Kober Integrals

2015 ◽  
Vol 18 (5) ◽  
Author(s):  
Moreno Concezzi ◽  
Roberto Garra ◽  
Renato Spigler
Keyword(s):  
Differential Equations ◽  
Numerical Approach ◽  
Numerical Results ◽  
Cauchy Type ◽  
Time Varying ◽  
Varying Coefficients ◽  
Fractional Relaxation ◽  
Time Varying Coefficients ◽  
Damped Oscillator

AbstractWe consider fractional relaxation and fractional oscillation equations involving Erdélyi-Kober integrals. In terms of the Riemann-Liouville integrals, the equations we analyze can be understood as equations with time-varying coefficients. Replacing the Riemann-Liouville integrals with Erdélyi-Kober-type integrals in certain fractional oscillation models, we obtain some more general integro-differential equations. The corresponding Cauchy-type problems can be solved numerically, and, in some cases analytically, in terms of the Saigo-Kilbas Mittag-Leffler functions. The numerical results are obtained by a treatment similar to that developed by K. Diethelm and N.J. Ford to solve the Bagley-Torvik equation. Novel results about the numerical approach to the fractional damped oscillator equation with time-varying coefficients are also presented.

Sign in / Sign up

Export Citation Format

Share Document